9 Problem B2.2 OGATA 4ed(Laplace Transform)

Laplace.T	Root-locus	Transient response	Frecuency response

9 Problem B2.2 OGATA 4ed(Laplace Transform)

Let us calculate the Laplace transform of the following functions.
$\begin{displaymath}f(t)=3\cdot sen(5\cdot t+\frac{\pi}{4})\end{displaymath}$

Solution:
We will decompose the function as follows:

$\begin{displaymath}f(t)=3\cdot sen(5\cdot t)\cdot cos(\frac{\pi}{4})+3\cdot cos(5\cdot t)\cdot sen(\frac{\pi}{4})\end{displaymath}$

We will use the following transforms

$\begin{displaymath}L(sen(w\cdot t))=\frac{w}{s^{2}+w^{2}}\end{displaymath}$

$\begin{displaymath}L(cos(w\cdot t))=\frac{s}{s^{2}+w^{2}}\end{displaymath}$

With what we get:

$\begin{displaymath}F(s)=3\cdot \frac{5}{s^{2}+5^{2}}\cdot cos(\frac{\pi}{4})+3\cdot \frac{s}{s^{2}+5^{2}}\cdot sen(\frac{\pi}{4})\end{displaymath}$
Let's check the result with Scilab..
t=0:0.5:50;
ft=3*sin(5*t+(%pi/4));
s=%s;
fs=3*(5/(s^2+5^2))*cos(%pi/4)+3*(s/(s^2+5^2))*sin(%pi/4);

fs2=syslin('c',fs);
fs1=csim('impulse',t,fs2);
clf;
subplot(2,1,1);
plot2d(t,ft,2);
xtitle('Phrasing');
xgrid;
subplot(2,1,2);
plot2d(t,fs1,1);
xtitle('Solution');
xgrid;
$\begin{displaymath}f(t)=0.03\cdot(1- cos(2\cdot t))\end{displaymath}$

Solucion:
We will decompose the function as follows:

$\begin{displaymath}f(t)=0.3- 0.3\cdot cos(2\cdot t)\end{displaymath}$

We will use the following transforms

$\begin{displaymath}L(1(t))=\frac{1}{s}\end{displaymath}$

$\begin{displaymath}L(cos(w\cdot t))=\frac{s}{s^{2}+w^{2}}\end{displaymath}$

With what we get:

$\begin{displaymath}F(s)=0.3\cdot \frac{1}{s}-0.3\cdot \frac{s}{s^{2}+2^{2}}\end{displaymath}$
Let's check the result with Scilab..
t=0:0.5:50;
ft=0.3*ones(size(t,2))-0.3*cos(2*t);
s=%s;
fs=0.3*(1/s)-0.3*(s/(s^2+2^2));
fs2=syslin('c',fs);
fs1=csim('impulse',t,fs2);
clf;
subplot(2,1,1);
plot2d(t,ft,2);
xtitle('Phrasing');
xgrid;
subplot(2,1,2);
plot2d(t,fs1,1);
xtitle('Solution');
xgrid;